Determine the angle between the lines y = 2x + 1 and y = -1/2x + 3. (2021)

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Q: Determine the angle between the lines y = 2x + 1 and y = -1/2x + 3. (2021)

Solution: The slopes are m1 = 2 and m2 = -1/2. The angle θ = tan⁻¹(|(m1 - m2) / (1 + m1*m2)|) = tan⁻¹(5/3), which is approximately 90 degrees.

Steps: 10

Step 1: Identify the equations of the lines. The first line is y = 2x + 1 and the second line is y = -1/2x + 3.
Step 2: Find the slope of the first line (m1). The slope is the coefficient of x, which is 2.
Step 3: Find the slope of the second line (m2). The slope is the coefficient of x, which is -1/2.
Step 4: Use the formula to find the angle θ between the two lines: θ = tan⁻¹(|(m1 - m2) / (1 + m1*m2)|).
Step 5: Calculate m1 - m2: 2 - (-1/2) = 2 + 1/2 = 2.5 or 5/2.
Step 6: Calculate m1 * m2: 2 * (-1/2) = -1.
Step 7: Substitute the values into the formula: θ = tan⁻¹(|(5/2) / (1 - 1)|).
Step 8: Since 1 + m1 * m2 = 1 - 1 = 0, we need to handle this case separately.
Step 9: Recognize that if the product of the slopes is -1, the lines are perpendicular, which means the angle is 90 degrees.
Step 10: Conclude that the angle between the lines is approximately 90 degrees.